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In many computer languages (such as FORTRAN or the Wolfram Language), the common residue of b (mod m) is written mod(b, m) (FORTRAN) or Mod[b, m] (Wolfram Language). The ...

Model theory is a general theory of interpretations of axiomatic set theory. It is the branch of logic studying mathematical structures by considering first-order sentences ...

The second-order ordinary differential equation x^2(d^2y)/(dx^2)+x(dy)/(dx)-(x^2+n^2)y=0. (1) The solutions are the modified Bessel functions of the first and second kinds, ...

The modified spherical Bessel differential equation is given by the spherical Bessel differential equation with a negative separation constant, ...

The group Gamma of all Möbius transformations of the form tau^'=(atau+b)/(ctau+d), (1) where a, b, c, and d are integers with ad-bc=1. The group can be represented by the 2×2 ...

A modular inverse of an integer b (modulo m) is the integer b^(-1) such that bb^(-1)=1 (mod m). A modular inverse can be computed in the Wolfram Language using PowerMod[b, ...

The direct sum of modules A and B is the module A direct sum B={a direct sum b|a in A,b in B}, (1) where all algebraic operations are defined componentwise. In particular, ...

Let a module M in an integral domain D_1 for R(sqrt(D)) be expressed using a two-element basis as M=[xi_1,xi_2], where xi_1 and xi_2 are in D_1. Then the different of the ...

The length of all composition series of a module M. According to the Jordan-Hölder theorem for modules, if M has any composition series, then all such series are equivalent. ...

Module multiplicity is a number associated with every nonzero finitely generated graded module M over a graded ring R for which the Hilbert series is defined. If dim(M)=d, ...